Diffraction by a Small Unstopped Lens

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Abstract
When monochromatic light is incident on a small converging lens which has no mount, the usual diffraction pattern is supplemented by light that has passed unobstructed around the lens. This problem arises in ice physics; very perfect lenses are formed by water occluded in the grain boundaries of polycrystalline ice, and measurement of the diffraction patterns they produce gives the ratio of the grain-boundary energy to that of an ice-water interface. The diffraction pattern is calculated here for the real region containing the focus, and also for the virtual region, because it is in the latter that the most sensitive measurements can be made. The intensity pattern depends entirely on a single parameter n, namely the number of half-period Fresnel zones subtended by the lens at its focus. The movement of the zeros of amplitude (wave dislocations) is followed as n changes. For discrete values of n an unstopped lens possesses a single anti-focus point, on axis in the virtual field, where the intensity is exactly zero.

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